### Arithmetical Properties of Laplacians of Graphs, by Dino J. Lorenzini

Let $M \in M_n (\mathbb Z)$ denote any matrix. Thinking
of $M$ as a linear map $M:{\mathbb Z}^n \rightarrow {\mathbb Z}^n$, we
denote by ${\Image}(M)$ the $\mathbb Z$-span of the column vectors
of $M$. Let $e_1, \dots, e_n,$ denote the standard basis
of ${\mathbb Z}^n$, and let $E_{ij} : = e_i - e_j$, $ (i \neq
j)$. In this article, we are interested in the group ${\mathbb Z}^n
/{\Image}(M)$, and in particular in the elements of this
group defined by the images $\tau_{ij}$ of the vectors $E_{ij}$ under the
quotient ${\mathbb Z}^n \rightarrow {\mathbb Z}^n / {\Image} (M)$.
Most of this article is devoted to the study of the case
where $M$ is the laplacian of a graph. In this case, the elements
$\tau_{ij}$ have finite order, and we study how the geometry of the graph
relates to these orders. Applications to the theory
of semistable reduction of curves will appear in a forthcoming article.

Dino J. Lorenzini <lorenzini@math.uga.edu>